cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A261064 a(n) = (3^n-1)*(n+1)/4.

Original entry on oeis.org

1, 6, 26, 100, 363, 1274, 4372, 14760, 49205, 162382, 531438, 1727180, 5580127, 17936130, 57395624, 182948560, 581130729, 1840247318, 5811307330, 18305618100, 57531942611, 180441092746, 564859072956, 1765184603000, 5507375961373, 17157594341214, 53379182394902
Offset: 1

Views

Author

R. J. Mathar, Aug 08 2015

Keywords

Comments

Second column of A201730.
Number of non-selfintersecting broken lines in a convex (n+1)-gon. (National Math Contest "Atanas Radev" 2020, Bulgaria) - Ivaylo Kortezov, Jan 18 2020

Links

Crossrefs

Cf. A001792, A027907, A201730, A212337.

Programs

  • Magma
    [(3^n-1)*(n+1)/4: n in [1..30]]; // Vincenzo Librandi, Aug 31 2016
  • Mathematica
    LinearRecurrence[{8, -22, 24, -9}, {1, 6, 26, 100}, 30] (* Vincenzo Librandi, Aug 31 2016 *)
Table[(3^n - 1)(n + 1)/4, {n, 0, 39}] (* Alonso del Arte, Jan 19 2020 *)
  • PARI
    first(m)=vector(m,i, (3^i-1)*(i+1)/4); /* Anders Hellström, Aug 08 2015 */

Formula

G.f.: -x*(-1 + 2*x) / ( (3*x - 1)^2*(x - 1)^2 ).
a(n) = A212337(n - 1) - 2*A212337(n - 2).
a(n) = Sum_{k = 1..n} A027907(n, 2k - 1)*k . - J. Conrad, Aug 30 2016
a(n) = Sum_{k = 0..(n - 1)} binomial(n + 1, k + 2)*A001792(k). - Ivaylo Kortezov, Jan 21 2020
E.g.f.: exp(x)*(exp(2*x)*(1 + 3*x) - x - 1)/4. - Stefano Spezia, May 14 2024

A081340 (5^n+(-1)^n)/2.

Original entry on oeis.org

1, 2, 13, 62, 313, 1562, 7813, 39062, 195313, 976562, 4882813, 24414062, 122070313, 610351562, 3051757813, 15258789062, 76293945313, 381469726562, 1907348632813, 9536743164062, 47683715820313, 238418579101562
Offset: 0

Views

Author

Paul Barry, Mar 18 2003

Keywords

Comments

Binomial transform of A003665. 2nd binomial transform of (1,0,9,0,81,0,729,0,..). Case k=2 of family of recurrences a(n)=2k*a(n-1)-(k^2-9)*a(n-2), a(0)=0, a(1)=k. A003665 is case k=1.

Links

Crossrefs

Cf. A081341, A087404.

Programs

Formula

a(n) = 4*a(n-1) + 5*a(n-2), a(0)=1, a(1)=2.
G.f.: (1-2*x)/((1+x)*(1-5*x)).
E.g.f.: exp(2*x) * cosh(3*x).
a(n) = ((2+sqrt(9))^n+(2-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
a(n) = sum( k=0..n, A201730(n,k)*8^k ). - Philippe Deléham, Dec 06 2011

A084132 a(n) = 4*a(n-1) + 6*a(n-2), a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 14, 68, 356, 1832, 9464, 48848, 252176, 1301792, 6720224, 34691648, 179087936, 924501632, 4772534144, 24637146368, 127183790336, 656558039552, 3389334900224, 17496687838208, 90322760754176, 466271170045952
Offset: 0

Views

Author

Paul Barry, May 16 2003

Keywords

Comments

Binomial transform of A002535.

Links

Crossrefs

Cf. A002535, A005667, A201730.

Programs

  • Magma
    [n le 2 select 2^(n-1) else 4*Self(n-1) +6*Self(n-2): n in [1..40]]; // G. C. Greubel, Oct 13 2022
  • Mathematica
    LinearRecurrence[{4,6}, {1,2}, 40] (* G. C. Greubel, Oct 13 2022 *)
  • SageMath
    [lucas_number2(n,4,-6)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009
  • SageMath
    A084132=BinaryRecurrenceSequence(4,6,1,2)
[A084132(n) for n in range(41)] # G. C. Greubel, Oct 13 2022

Formula

a(n) = (2+sqrt(10))^n/2 + (2-sqrt(10))^n/2.
G.f.: (1-2*x)/(1-4*x-6*x^2).
E.g.f.: exp(2*x)*cosh(sqrt(10)*x).
a(n) = Sum_{k=0..n} A201730(n,k)*9^k. - Philippe Deléham, Dec 06 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(5*k-2)/(x*(5*k+3) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
a(n) = 2*i^n*6^((n-2)/2)*( 3*ChebyshevU(n, 2/(i*sqrt(6))) + i*sqrt(6)*ChebyshevU(n -1, 2/(i*sqrt(6))) ). - G. C. Greubel, Oct 13 2022
Previous Showing 11-13 of 13 results.

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